Random assignment problems

Krokhmal, Pavlo A.; Pardalos, Pãnos M.

doi:10.1016/j.ejor.2007.11.062

Cited by 51 publications

(35 citation statements)

References 94 publications

(179 reference statements)

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“…For the assignment problem, such results were proposed by computational experiments and then proven theoretically. For a survey on the so-called "random assignment problem" and several of its generalizations, see [Krokhmal and Pardalos, 2009]. In particular, a result for the assignment problem on random instances in complete bipartite graphs and i. i. d. uniform random variables on [0, 1] or i. i. d. exponential random variables with mean 1 as edge costs can be proven.…”

Section: Chapter 4 Optimal Solutions Of Random Instances With Standarmentioning

confidence: 99%

“…The polytope of the standard integer linear programming formulation for the assignment problem is well-understood. For random instances, expected values and other results are known for certain cost functions [Krokhmal and Pardalos, 2009]. Also, several, often N P -complete, generalizations of the assignment problem such as the quadratic and the multixii dimensional assignment problem have been investigated.…”

Section: Prefacementioning

confidence: 99%

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The hypergraph assignment problem

Borndörfer

Heismann

2015

Discrete Optimization

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Section: Chapter 4 Optimal Solutions Of Random Instances With Standarmentioning

confidence: 99%

Section: Prefacementioning

confidence: 99%

The hypergraph assignment problem

Borndörfer

Heismann

2015

Discrete Optimization

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“…However, in our case it is difficult to solve (6) because enthalpy h i,j is a nonlinear function of pressure P j due to thermobaricity. Fortunately, our problem (6) is actually the famous ''assignment problem'' in computational mathematics (Derigs 1985;Martello and Toth 1987;Bertsekas 1988;Martello et al 2000;Burkard et al 2009;Krokhmal and Pardalos 2009). This problem was effectively solved by the Hungarian algorithm (HA) with 100% accuracy (Kuhn 1955;Lawler 1976;Burkard et al 2009).…”

Section: Calculation Strategy For Ocapementioning

confidence: 99%

Ocean Convective Available Potential Energy. Part I: Concept and Calculation

Ingersoll

Stewart

et al. 2016

Journal of Physical Oceanography

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Thermobaric convection (type II convection) and thermobaric cabbeling (type III convection) might substantially contribute to vertical mixing, vertical heat transport, and deep-water formation in the World Ocean. However, the extent of this contribution remains poorly constrained. The concept of ocean convective available potential energy (OCAPE), the thermobaric energy source for type II and type III convection, is introduced to improve the diagnosis and prediction of these convection events. OCAPE is analogous to atmospheric CAPE, which is a key energy source for atmospheric moist convection and has long been used to forecast moist convection. OCAPE is the potential energy (PE) stored in an ocean column arising from thermobaricity, defined as the difference between the PE of the ocean column and its minimum possible PE under adiabatic vertical parcel rearrangements. An ocean column may be stably stratified and still have nonzero OCAPE. The authors present an efficient strategy for computing OCAPE accurately for any given column of seawater. They further derive analytical expressions for OCAPE for approximately two-layer ocean columns that are widely observed in polar oceans. This elucidates the dependence of OCAPE on key physical parameters. Hydrographic profiles from the winter Weddell Sea are shown to contain OCAPE (0.001-0.01 J kg 21), and scaling analysis suggests that OCAPE may be substantially enhanced by wintertime surface buoyancy loss. The release of this OCAPE may substantially contribute to the kinetic energy of deep convection in polar oceans.

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“…Such a random matrix could be either an unitary matrix [9], the product with its own transpose matrix being equal to the identity matrix, or simply consisting of completely independent random variables. Likewise Random Assignment problem [12][13][14] and randomly generated cities for TSP [15,16], or random graph theory [17,18] are considered and studied.…”

Section: Introductionmentioning

confidence: 99%

Machine Learning Meliorates Computing and Robustness in Discrete Combinatorial Optimization Problems

Hsieh

Fujii

Hsieh

2016

Front. Appl. Math. Stat.

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Discrete combinatorial optimization problems in the real world are typically defined via an ensemble of potentially high dimensional measurements pertaining to all subjects of a system under study. We point out that such a data ensemble in fact embeds with system's information content that is not directly used in defining the combinatorial optimization problems. Can machine learning algorithms extract such information content and make combinatorial optimizing tasks more efficient? Would such algorithmic computations bring new perspectives into this classic topic of Applied Mathematics and Theoretical Computer Science? We show that answers to both questions are positive. One key reason is due to permutation invariance. That is, the data ensemble of subjects' measurement vectors is permutation invariant when it is represented through a subject-vs.-measurement matrix. An unsupervised machine learning algorithm, called Data Mechanics (DM), is applied to find optimal permutations on row and column axes such that the permuted matrix reveals coupled deterministic and stochastic structures as the system's information content. The deterministic structures are shown to facilitate geometry-based divide-and-conquer scheme that helps optimizing task, while stochastic structures are used to generate an ensemble of mimicries retaining the deterministic structures, and then reveal the robustness pertaining to the original version of optimal solution. Two simulated systems, Assignment problem and Traveling Salesman problem, are considered. Beyond demonstrating computational advantages and intrinsic robustness in the two systems, we propose brand new robust optimal solutions. We believe such robust versions of optimal solutions are potentially more realistic and practical in real world settings.

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Random assignment problems

Cited by 51 publications

References 94 publications

The hypergraph assignment problem

The hypergraph assignment problem

Ocean Convective Available Potential Energy. Part I: Concept and Calculation

Machine Learning Meliorates Computing and Robustness in Discrete Combinatorial Optimization Problems

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